I've heard people say both. Which is the right one?
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This is the right one. Before superlative adjectives you should use "the".
I do not find "Produce best output" normal in any dialect of English that I recognise.
As Manoochehr says, superlatives usually need "the", but there is an idiom "for best results" (typically on instruction sheets and clothing labels), and I wonder if somebody has tried to generalise that construction.
I can't see that "try and" has any relevance.